Polychromatic X-ray tomography: direct quantitative phase reconstruction

doi:10.1364/OE.20.023361

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Polychromatic X-ray tomography: direct quantitative phase reconstruction

Benedicta D. Arhatari, Grant van Riessen, and Andrew Peele »View Author Affiliations

Optics Express, Vol. 20, Issue 21, pp. 23361-23366 (2012)
http://dx.doi.org/10.1364/OE.20.023361

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– Abstract
1. Introduction
2. Reconstruction algorithm
3. Experimental results
4. Conclusion
– References and links

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Abstract

We describe a direct quantitative phase reconstruction approach using an X-ray laboratory-based source. Using a single phase-contrast image from each tomographic projection we show that it is possible to modify the filter term in a filtered back projection reconstruction to take account of the broad spectrum from a laboratory source. The accessibility of conventional X-ray laboratory sources makes this method very useful for quantitative phase imaging of homogeneous and weakly absorbing objects.

1. Introduction

Phase contrast is a class of imaging that has been developed since the 1930s by Zernike [1

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942). [CrossRef]

] to improve visualization of transparent objects in optical microscopy. Development of X-ray phase contrast began in the 1980s [2

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980). [CrossRef]

] for imaging objects with low electron density, such as low Z and low density materials. Phase retrieval in propagation based x-ray imaging is the inversion of a phase contrast image to quantitatively obtain the phase of a complex wave field at the exit surface plane of a sample, from which the decrement of the real part of the refractive index from unity of a sample is often referred, typically using the projection approximation [3

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).

]. This component of the sample's refractive index is referred to here as the phase of the sample.

The phase of sample recovered by phase retrieval is the linear projection of the sample phase through a given projection of that sample. Accordingly, standard filtered back projection methods can be used to reconstruct the 3D distribution of the phase of the sample [4

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003). [CrossRef] [PubMed]

]. Under certain conditions it has been demonstrated that the phase retrieval step can be incorporated as part of the filter in filtered back projection to obtain the 3D distribution of the phase of the sample in a single step [5

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002). [CrossRef] [PubMed]

]. These conditions have been extended from a requirement that the sample be non-absorbing [5

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002). [CrossRef] [PubMed]

], to cases where the sample is weakly absorbing using a semi-empirical approach [6

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14(18), 8103–8110 (2006). [CrossRef] [PubMed]

] or knowledge of the sample composition [7

B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007). [CrossRef] [PubMed]

]. Algorithms based on the contrast transfer function approach [8

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

] or which use the phase-attenuation duality [9

R. C. Chen, H. L. Xie, L. Rigon, R. Longo, E. Castelli, and T. Q. Xiao, “Phase retrieval in quantitative x-ray microtomography with a single sample-to-detector distance,” Opt. Lett. 36(9), 1719–1721 (2011). [CrossRef] [PubMed]

] have also been demonstrated.

These approaches have typically been implemented using monochromatic light from a synchrotron source. However, it has long been a feature of propagation based phase contrast methods that the broad band spectrum delivered by a laboratory source can be used where the source size is sufficiently small [10

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]

, 11

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004). [CrossRef]

]. Furthermore, it has been demonstrated that quantitative phase retrieval can be performed using a laboratory source [12

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008). [CrossRef] [PubMed]

]. This approach uses spectrally weighted values for the material properties corresponding to the sample phase and attenuation [12

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008). [CrossRef] [PubMed]

]. In this paper, we demonstrate that a similar approach can be used to incorporate the phase retrieval step within the tomographic reconstruction process. This provides a single-step process for the quantitative phase retrieved reconstruction of a three-dimensional complex refractive index distribution.

2. Reconstruction algorithm

In the Transport of Intensity approximation [13

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996). [CrossRef] [PubMed]

] for an homogeneous object (one that consists of a single material) that is weakly absorbing it is possible to express the monochromatic intensity, I, at coordinate (x,y) for a given projection, θ, measured in a plane at distance, z, from the sample, assuming that the phase of the unpropagated field is sufficiently slowly varying in x and y, as [3

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).

, 7

I_{θ, λ}^{z} (x, y) = I_{0}^{} [1 + g_{θ, λ} (x, y)] .

(1)

Here λ is the monochromatic wavelength,

I_{0}

is the incident intensity and

g_{θ, λ} (x, y) = - μ_{λ} {\hat{t}}_{θ} (x, y) + z δ_{λ} \nabla^{2} {\hat{t}}_{θ} (x, y),

(2)

where

\nabla^{2} = (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}})

and for a nominal density: µ_λ is the linear absorption coefficient and δ_λ is the decrement from unity of the real part of the refractive index; n_λ = 1-δ_λ + iβ_λ, with

μ_{λ} = \frac{4 π β_{λ}}{λ}

. Also,

{\hat{t}}_{θ} (x, y) = \int_{s a m p l e_{θ}} ρ_{f} (x, y, z) d z

(3)

is a projection operator which gives the linear integral of the density fraction, ρ_f, through the thickness of the sample for a given projection angle, θ. The density fraction is the factor that scales the nominal density to the actual density at a given location in the sample. If ρ_f = 1 then the projection operator returns the thickness of the sample.

To extend Eq. (1) to take account of a polychromatic source we note that the measured intensity is the integral over the source spectrum of the monochromatic intensity as modified by the detector response and any wavelength dependent effects in the optical path, such as air scatter or absorption, which can be represented by the combined function, D_λ. The D_λ function for the experimental set-up used in this paper is shown in Fig. 1 . as dotted lines. Accordingly,

\frac{\int I_{θ, λ}^{z} (x, y) D_{λ} d λ}{\int D_{λ} d λ} = \frac{\int I_{0}^{} [1 + g_{θ, λ} (x, y)] D_{λ} d λ}{\int D_{λ} d λ}

(4)

This can be written as:

I_{θ, p o l y}^{z} (x, y) = I_{0}^{} [1 + g_{θ, p o l y} (x, y)] .

(5)

Using the same approach we can also express g_θ,λ in Eq. (2) in terms of the effective polychromatic materials coefficients,

g_{θ, p o l y} (x, y) = - μ_{p o l y} {\hat{t}}_{θ} (x, y) + z δ_{p o l y} \nabla^{2} {\hat{t}}_{θ} (x, y)

(6)

where µ_poly is the polychromatic case of the linear absorption coefficient defined by the spectrally weighted value of µ_λ with the D_λ function (as shown in Eqs. (4) and (5) for the intensity) and δ_poly is defined similarly as the polychromatic counter part of δ_λ. This expression confines the wavelength-dependent terms to expressions that can be calculated knowing the composition of the sample.

Fig. 1 The linear attenuation of polystyrene as a function of energy and the calculated effective value µ_poly for the measured source spectrum is shown in (a). Similarly δ and δ_poly are shown on (b). The combined detector response function and X-ray source spectrum for a Tungsten target at 40kV are shown in both Fig (a) and (b) (dotted lines).

Following earlier work [5

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002). [CrossRef] [PubMed]

] we have previously shown [7

] that an expression in the form of Eq. (6) can be combined with the approach used in the filtered back projection inversion of the Radon transform to perform a phase retrieved 3D reconstruction of the density fraction distribution throughout the object. This results in the following expression:

ρ_{f} (x, y, z) = - \frac{1}{δ_{p o l y}} \int_{0}^{π} q_{θ} (x \cos θ + z \sin θ, y) d θ,

(7)

where

q_{θ} = F^{- 1} [S . G_{θ, p o l y}]

. Here

F^{- 1}

is the inverse Fourier transform operator and G_θ,poly is the Fourier transform of g_θ,poly, which can be obtained from measurement via Eq. (5), and

S (ξ, η) = \frac{| ξ |}{\frac{μ_{p o l y}}{δ_{p o l y}} + 4 π^{2} z (ξ^{2} + η^{2})},

(8)

where ξ, η are the Fourier conjugate variables for x and y respectively. For a pure phase object (when µ_poly = 0), Eq. (8) becomes identical to the one derived by Bronnikov [5

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002). [CrossRef] [PubMed]

, 14

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]

]. As with the earlier monochromatic approach [7

] Eq. (8) has the advantage of stability due to the denominator having no zero values. The benefit of this approach is that for a weakly absorbing homogeneous object in the TIE regime the density distribution throughout a sample can be obtained using a single reconstruction step by obtaining a single measurement of intensity at each projection.

3. Experimental results

To demonstrate our method polystyrene (C₉H₁₂) micro-spheres of 74µm diameter with a nominal density of 1.05g/cm³ were imaged using a laboratory-based micro X-ray Computed Tomography system (Xradia Inc.). The X-ray source was a closed X-ray tube containing Tungsten as a target material and operated with a tube voltage of 40kV. In this energy, the x-ray source has the size of about 6µm. The spectrum of the source was measured using an energy sensitive detector (XR-100T-CdTe, AMPTEK Inc.). Figure 1 shows the μ and δ values for polystyrene in the same energy range of the measured source spectrum. The absorption and phase coefficients, μ_poly and δ_poly, were calculated using the spectral weighting distribution used in Eq. (4) that is also shown in Fig. 1. In this case, μ_poly is 140.6 m⁻¹, which is equal to the linear attenuation at 10.9keV (μ_10.9keV) and δ_poly is 1.69x10⁻⁶ (δ_11.9keV). The calculated values of μ_poly and δ_poly correspond to different energies. This is to be expected as μ and δ have a different energy dependency of E⁻³ and E⁻² respectively [15

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000). [CrossRef]

]. This highlights why a very broad-band spectrum cannot be understood as a single effective energy whereas in the narrow band approach [16

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007). [CrossRef]

] we can use one effective energy.

The sample was placed at a distance of 70 mm from the source and 83.5 mm from the imaging detector. Based on the Fresnel scaling theorem [3

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).

], this is equivalent to an effective propagation distance of 38mm. The imaging detector used was different from the energy sensitive detector mentioned above. The imaging detector comprised of a 250µm CsI(Tl) scintillator coupled to a CCD camera by a 4 × objective lens. The CCD camera (Andor Technology) has 2048 × 2048 pixels with a physical pixel size of 13.5 µm. 2 × binning was used in the data acquisition to reduce the exposure time. The total image magnification from this set-up is 8.8, and an effective pixel size is 3.03 μm.

An image of the polystyrene sample for one projection is shown in Fig. 2(a) . The image contains 512 × 120 pixels. Each image was corrected for the dark current and for the non-uniform illumination in the imaging system, determined by taking a background image of the beam without the sample present. The edge enhancement typical of phase contrast imaging is clearly visible as fringes on the edge of the spheres. Each projection image was recorded in 10s. In order to collect a three-dimensional data set, the sample was scanned by acquiring 451 projections at equal angles through an angular range of 180°. The distribution of the term from Eq. (8) is shown in Fig. 2(b). We used μ_poly 140.6 m⁻¹, δ_poly 1.69x10⁻⁶ and z 38mm to calculated Eq. (8).

Fig. 2 (a). The recorded intensity image of polystyrene micro-spheres using a polychromatic X-ray source at 40kV. (b). The distribution of Eq. (8) for polystyrene using the set-up described in the text.

Using the acquired projection data sets, we then performed the filtered back projection tomographic reconstruction based on Eq. (7) using the filter shown in Fig. 2(b). The 3D reconstruction result is shown in Fig. 3(a) . Figure 3(b) shows a slice of 512 × 512 pixels from the reconstruction with the corresponding plot in Fig. 3(c) along the dashed horizontal line in Fig. 3(b). The plot shows the quantitative value of the function ρ_f(x,y,z). The retrieved values of 1 ± 0.05 in the area around the center of the spheres are slightly below from the expected value of ρ_f = 1. The dimension of the spheres can be measured determined by the width of the features in Fig. 3(c) using the known magnification. It is in agreement with the manufacture's specified diameter of 74 ± 11.8µm.

Fig. 3 (a) The rendering based on the reconstruction result of polystyrene micro-spheres using Eq. (7). (b). One of the reconstructed slices. (c). The corresponding plot along the dash line shown in (b).

In Fig. 4(a) , we compare the results obtained by choosing a single effective energy and using the corresponding values for μ and δ. An effective energy of 16.2 keV was calculated by making a weighted sum of the energy spectrum. The density fraction calculated using the effective energy is shown with a dotted line. Another comparison is also made by using the energy of the maximum spectral intensity (8.4 keV). The density fraction calculated using this energy is presented as a dashed line. From the plot it is obvious that both the peak and the effective energy approach do not yield satisfactory results. Figures 4(b) and 4(c) show the capability of the method to handle a non-homogeneous sample using the same set-up. Polyimide film with a density of 1.45g/cm³ is inserted around the polystyrene spheres. Polyimide (Kapton, C₂₂H₁₀N₂O₄) has the value of μ_poly 285.1 m⁻¹ and δ_poly 2.19x10⁻⁶ in this energy range. The same filter using the polystyrene parameter shown above in Fig. 2(b) was been used for the reconstruction process. The plot in Fig. 4(c) shows that the polystyrene spheres were reconstructed with the correct density value while this is not the case for the polyimide film. The second peak from the left in the plot originated from the kapton film and its density fraction deviates from the expected value of 1. However, the reconstruction slice in Fig. 4(b) still shows a good qualitative result. The breach of the weakly absorbing and slowly varying phase requirement can be seen in Fig. 4(b) as resulting in an artefact in the elongated direction of the kapton film. This is due to the fact that the film is no longer weakly interacting with x-rays in some angles of projection.

Fig. 4 (a) Comparison of the nominal bulk density distribution of polystyrene spheres determined by different approaches. The solid line is calculated using Eq. (7) and 8), the dotted line is obtained by using an effective energy, and the dashed line is obtained by using an energy corresponding to maximum intensity in the source spectrum. (b). The reconstruction slice of a combination of polyimide film and polystyrene spheres. (c). The plot along the dashed horizontal line in Fig. 4(b).

Without a priori knowledge of the sample, our method still behaves well in the sense that there are no artefacts produced when a weakly absorbing sample is used. However the value will not be quantitative.

4. Conclusion

We have demonstrated a direct reconstruction algorithm for quantitative phase tomographic imaging using an extremely broad source spectrum for homogeneous and weakly absorbing samples. We need only a single distance projection data set collected from a laboratory polychromatic X-ray source. This method may find widespread application for quantitative phase imaging since it makes efficient use of X-ray laboratory sources that are more widely accessible than monochromatic synchrotron X-ray sources.

Acknowledgments

The authors acknowledge the support of the Australian Research Council through the Centre of Excellence for Coherent X-ray Science.

References and links

1.	F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942). [CrossRef]
2.	E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980). [CrossRef]
3.	D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
4.	S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003). [CrossRef] [PubMed]
5.	A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002). [CrossRef] [PubMed]
6.	A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14(18), 8103–8110 (2006). [CrossRef] [PubMed]
7.	B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007). [CrossRef] [PubMed]
8.	T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]
9.	R. C. Chen, H. L. Xie, L. Rigon, R. Longo, E. Castelli, and T. Q. Xiao, “Phase retrieval in quantitative x-ray microtomography with a single sample-to-detector distance,” Opt. Lett. 36(9), 1719–1721 (2011). [CrossRef] [PubMed]
10.	S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]
11.	B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004). [CrossRef]
12.	B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008). [CrossRef] [PubMed]
13.	K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996). [CrossRef] [PubMed]
14.	A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]
15.	R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000). [CrossRef]
16.	G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007). [CrossRef]

OCIS Codes
(110.6960) Imaging systems : Tomography
(340.0340) X-ray optics : X-ray optics
(340.7440) X-ray optics : X-ray imaging

ToC Category:
X-ray Optics

History
Original Manuscript: July 17, 2012
Revised Manuscript: September 21, 2012
Manuscript Accepted: September 21, 2012
Published: September 26, 2012

Citation
Benedicta D. Arhatari, Grant van Riessen, and Andrew Peele, "Polychromatic X-ray tomography: direct quantitative phase reconstruction," Opt. Express 20, 23361-23366 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23361

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References

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9(7), 686–698 (1942). [CrossRef]
E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech.15(8), 937–945 (1980). [CrossRef]
D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11(19), 2289–2302 (2003). [CrossRef] [PubMed]
A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A19(3), 472–480 (2002). [CrossRef] [PubMed]
A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express14(18), 8103–8110 (2006). [CrossRef] [PubMed]
B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum.78(5), 053701 (2007). [CrossRef] [PubMed]
T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006). [CrossRef]
R. C. Chen, H. L. Xie, L. Rigon, R. Longo, E. Castelli, and T. Q. Xiao, “Phase retrieval in quantitative x-ray microtomography with a single sample-to-detector distance,” Opt. Lett.36(9), 1719–1721 (2011). [CrossRef] [PubMed]
S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996). [CrossRef]
B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum.75(12), 5271–5276 (2004). [CrossRef]
B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express16(24), 19950–19956 (2008). [CrossRef] [PubMed]
K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996). [CrossRef] [PubMed]
A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun.171(4-6), 239–244 (1999). [CrossRef]
R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today53(7), 23–26 (2000). [CrossRef]
G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007). [CrossRef]

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